Problem: Rewrite as a simplified fraction. ${1.65\overline{7} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1657.7777...\\ 100x &= 165.7777...\end{align*}}$ Subtracting the two equations, we have: ${900x = 1492}$ ${x = \dfrac{1492}{900}} $ ${x = \dfrac{373}{225}} \text{or} {1 \dfrac{148}{225}}$